Nonlinear Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic, cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves i

Existence And Stability Of High Frequency Standing Waves For A Nonlinear Schrodinger Equation

François Samer Genoud

This article is concerned with the existence and orbital stability of standing waves for a nonlinear Schrodinger equation (NLS) with a nonautonomous nonlinearity. It continues and concludes the series of papers [6, 7, 8]. In [6], the authors make use of a continuation argument to establish the existence in R x H-1(R-N) of a smooth local branch of solutions to the stationary elliptic problem associated with (NLS) and hence the existence of standing wave solutions of (NLS) with small frequencies. Complementary conditions on the nonlinearity are found, under which either stability of the standing waves and bifurcation of the branch of solutions from the point (0, 0) is an element of R x H-1 (R-N) occur, or instability and asymptotic bifurcation occur. The main hypotheses in [6] concern the behaviour of the nonlinearity with respect to the space variable at infinity. The paper [7] extends the results of [6] to (NLS) with more general nonlinearities. In [8], the global continuation of the local branch obtained in [6] is proved under additional hypotheses on the nonlinearity. In particular, spherical symmetry with respect to the space variable is assumed. The aim of the present work is to prove the existence and discuss the orbital stability of standing waves with high frequencies, independently of the results obtained in [6] and [8]. The main hypotheses now concern the behaviour of the nonlinearity with respect to the space variable around the origin. The methods are the same in spirit as that of [6] and permit to discuss the asymptotic behaviour of the global branch of solutions obtained in [8].

2009

Efficient geometric integrators for the linear and nonlinear time-dependent Schrödinger equation

Julien Roulet

Many physical and chemical reactions are driven by nonadiabatic processes, which imply the breakdown of the celebrated Born-Oppenheimer approximation. To understand these processes, experimentalists employ spectroscopic techniques. However, the obtained results are difficult to decipher, and accurate molecular quantum dynamics simulations are used to interpret the results.The second-order split-operator algorithm is one of the most popular numerical methods for simulating the nonadiabatic quantum dynamics because it is explicit, easy to implement, and it preserves many geometric properties of the exact solution. However, the second-order accuracy of this algorithm makes it unaffordable if very accurate results are needed, as tiny time steps are required. To remedy this lack of efficiency, we use composition methods to generate higher-order split-operator algorithms.Although compositions methods increase the accuracy of the standard split-operator algorithm to arbitrary even orders of convergence, the efficiency of the obtained algorithms is still questioned because the computational cost per time step increases drastically with the order of convergence. Therefore, using one- and three-dimensional models of NaI and pyrazine, respectively, we investigate the convergence, efficiency, and geometric properties of these high-order integrators and find that they are, for accurate simulations, more efficient than the standard split-operator algorithm while still preserving the same geometric properties. Besides employing these integrators for simulating the nonadiabatic quantum dynamics, we also explore quantum control and, more specifically, local control theory. This technique uses the instantaneous dynamics of the system to compute an electric field, which interacts with the system in order to drive the state in a desired direction. Because the electric field is obtained from the state itself, we demonstrate that this technique translates into a nonlinear time-dependent Schrödinger equation. Although it is geometric and second-order accurate for simple nonlinearities, the standard split-operator algorithm loses its time-reversal symmetry and second-order accuracy when employed for more complicated nonlinear time-dependent Schrödinger equations. One example of the latter is the one appearing in local control theory.We demonstrate that this lack of generality of the standard split-operator algorithm occurs due to its explicit nature. Thus, we propose two strategies to overcome this issue: First, we completely abandon the split-operator algorithm and present a numerical method based on the implicit midpoint method instead. Second, we make the standard split-operator algorithm implicit, which avoids abandoning the split-operator algorithm altogether. The accuracy and geometric properties of both strategies are then numerically verified using a two-dimensional model of retinal, a molecule whose photochemistry triggers the first event in the biological process of vision. The results demonstrate that both approaches yield second-order methods that preserve all the geometric properties of the exact solution. Because the developed integrators are symmetric, we further improve their accuracy and efficiency using composition methods.

EPFL2022

Efficient geometric integrators for the linear and nonlinear time-dependent Schrödinger equation

Julien Roulet

EPFL2022